On topological entropy of commuting tree maps

Let T be a tree with s ends and f,g be continuous maps from T to T with f∘g=g∘f. In this note we show that if there exists a positive integer m≥2 such that gcd(m,l)=1 for any 2≤l≤s and f,g share a periodic point which is a km-periodic point of f for some positive integer k, then the topological entropy of f∘g is positive.